laplace transform of f(t) = t sin (at) use integration by parts
To find the Laplace transform of the function f(t) = t sin (at) using integration by parts, follow these steps:
Step 1: Write down the given function.
f(t) = t sin (at)
Step 2: Recall the formula for integration by parts:
∫u dv = uv - ∫v du
Step 3: Select u and dv:
Let u = t (choose the simplest function)
Let dv = sin (at) dt
Step 4: Differentiate u to get du:
du/dt = 1
Step 5: Integrate dv to get v:
To integrate sin (at) dt, divide both sides by a to make it simpler:
dv = (1/a) sin (at) dt
Integrate using the substitution method:
Let x = at, then dx = a dt
dv = (1/a) sin (at) dt
= (1/a) sin (x) dx
= - (1/a) cos (x) + C1
Now, substitute x = at back in:
v = - (1/a) cos (at) + C1
Step 6: Apply the integration by parts formula:
∫u dv = uv - ∫v du
∫(t sin (at)) dt = (- (1/a) t cos (at) + C1) - ∫[(- (1/a) cos (at) + C1)] dt
Step 7: Simplify the equation:
∫(t sin (at)) dt = - (1/a) t cos (at) + C1 + (∫(1/a) cos (at) dt - ∫C1 dt)
∫(t sin (at)) dt = - (1/a) t cos (at) + C1 + (1/a) ∫cos (at) dt - ∫C1 dt
Note that ∫C1 dt is simply C1t
∫(t sin (at)) dt = - (1/a) t cos (at) + C1 + (1/a^2) sin (at) - C1t + C2
Step 8: Simplify further:
∫(t sin (at)) dt = - (1/a) t cos (at) + (1/a^2) sin (at) - C1t + (C1 + C2)
Step 9: Rename constants:
Let C = C1 + C2
Step 10: Final expression:
The Laplace transform of f(t) = t sin (at) is given by:
L{f(t)} = - (1/a) L{t cos (at)} + (1/a^2) L{sin (at)} - C1 L{t} + C
Note: You can find the Laplace transforms of each term separately using standard Laplace transformation tables or properties.